Question

(a) find the common ratio of the geometric series, \((b)\) write the function that gives the sum of the series, and (c) use a graphing utility to graph the function and the partial sums \(S_{3}\) and \(S_{5} .\) What do you notice? $$ 1-\frac{x}{2}+\frac{x^{2}}{4}-\frac{x^{3}}{8}+\cdots $$

Short answer

The common ratio for the geometric series is \( -\frac{x}{2} \). The sum of series function is \( S = \frac{2}{2 + x} \). The graph of the function and the partial sums \( S_{3} \) and \( S_{5} \) shows convergence of the series with increasing number of terms.

Step-by-step solution

Find the Common Ratio

The common ratio \( r \) of a geometric series can be found by dividing any term in the series by the preceding term. Let's use the first two terms of the series. So, \( r = -\frac{x/2}{1} = -\frac{x}{2} \)

Write the Sum Function

Now, with the common ratio and the first term of the series, the sum of the series can be written as \( S = \frac{a}{1 - r} \). Using the values \( a = 1 \) (the first term of the series) and \( r = -x/2 \), we substitute to get the sum equation which is \( S = \frac{1}{1 - (-x/2)} = \frac{1}{1 + \frac{x}{2}} = \frac{2}{2 + x} \).

Graph the Function and the Partial Sums

Use a graphing tool to plot the function \( \frac{2}{2 + x} \) and the partial sums \( S_{3} \) and \( S_{5} \). \( S_{3} \) is the sum of the first three terms of the series and \( S_{5} \) is the sum of the first five terms. Calculate these partial sums from the series terms. Then plot \( S, S_{3}, \) and \( S_{5} \).